2006 iTest Problems/Problem 36

Let $\alpha$ denote $\cos^{-1}(\tfrac 23)$ . The recursive sequence $a_0,a_1,a_2,\ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$ , $a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$ Suppose that the series $\sum_{k=0}^\infty\dfrac{a_k}{2^k}$ can be expressed uniquely as $\tfrac{p\sqrt q}r$ , where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$ .

Solution

We write $\sum_{k=0}^n a_k a_{n-k} = \cos(n\alpha)$ by rearranging the defining equation and using $a_0 = 1$ . Summing this with weights $\frac{1}{2^n}$ from zero to infinity, we get $\sum_{n=0}^{\infty}\frac{1}{2^n}\sum_{n}^{k=0} a_k a_{n-k} = \sum_{n=0}^{\infty} \frac{\cos(n\alpha)}{2^n}$ . We can rewrite this as $(\sum_{n=0}^{\infty} \frac{a_n}{2^n})^2 = \sum_{n=0}^{\infty} \frac{\cos(n\alpha)}{2^n}$ .

Next, we compute $\sum_{n=0}^{\infty} \frac{\cos(n\alpha)}{2^n} = \frac{1}{2}(\sum_{n=0}^{\infty} \frac{e^{in\alpha}}{2^n} + \sum_{n=0}^{\infty} \frac{e^{-in\alpha}}{2^n}) = \frac{1}{2}(\frac{1}{1-\frac{e^{i\alpha}}{2}} + \frac{1}{1-\frac{e^{-i\alpha}}{2}})$ , which simplifies to $\frac{1 - \frac{e^{i\alpha} + e^{-i\alpha}}{4}}{\frac{5}{4} - \frac{e^{i\alpha} + e^{-i\alpha}}{2}}$ . Since $\frac{e^{i\alpha} + e^{-i\alpha}}{2} = \cos\alpha = \frac{2}{3}$ , the entire expression becomes $\frac{1 - \frac{1}{2}\cdot\frac{2}{3}}{\frac{5}{4} - \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{7}{12}} = \frac{8}{7}$ .

Taking square roots, we get $\sum_{n=0}^{\infty} \frac{a_n}{2^n} = \sqrt{\frac{8}{7}} = \frac{2\sqrt{14}}{7}$ , so our answer is $2 + 14 + 7 = \boxed{23}$ and we are done.

2006 iTest (Problems, Answer Key)
Preceded by: Problem 35	Followed by: Problem 37
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2006 iTest Problems/Problem 36

Solution

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